Note on the Solvability of Equations Involving Unbounded Linear and Quasibounded Nonlinear Operators*

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 56, 495-501 (1976)

W. V. PETRYSHYN

Deportment of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Submitted by Ky Fan

INTRODUCTION

Let X and Y be Banach spaces, X* and Y* their respective duals and (u, x) the value of u in X* at x in X. For a bounded linear operator T: X---f Y (i.e., for T EL(X, Y)) let N(T) C X and R(T) C Y denote the null space and the range of T respectively and let T *: Y* --f X* denote the adjoint of T. Foranyset vinXandWinX*weset I”-={uEX*~(U,X) =OV~EV) and WL = {x E X 1(u, x) = OVu E W). In [6] Kachurovskii obtained (without proof) a partial generalization of the closed range theorem of Banach for mildly nonlinear equations which can be stated as follows:

THEOREM K. Let T EL(X, Y) have a closed range R(T) and a Jinite dimensional null space N(T). Let S: X --f Y be a nonlinear compact mapping such that R(S) C N( T*)l and S is asymptotically zer0.l Then the equation TX + S(x) = y is solvable for a given y in Y if and only if y E N( T*)l. The purpose of this note is to extend Theorem K to equations of the form

TX + S(x) = y (x E WY, y E Y), (1) where T: D(T) X--f Y is a closed linear operator with domain D(T) not necessarily dense in X and such that R(T) is closed and N(T) (CD(T)) has a closed complementary subspace in X and S: X-t Y is a nonlinear quasibounded mapping but not necessarily compact or asymptotically zero. Our extension, Theorem 1 below, is such that it also includes as special cases Theorem 3 of Kachurovskii [6] as well as the surjectivity theorems of Granas [4], George [3], Vignoli [15], and Petryshyn [12].

* Supported in part by the NSF Grant GP-20228 and in part by the Research Council of Rutgers University while the author was on the faculty research fellowship during the academic year 1974-75. 1 See Section 1 for the various notions mentioned in the Introduction. 495 Copyright $3 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. 496 w. v. PETRYSHYN

The proof of our Theorem I is based on certain results of Kato [7] and the author’s fixed point Theorem 1 in [13] for l-set-contractions.

1. THE MAIN THEOREWI

To state our result we must first introduce certain notations and definitions. For a general closed linear operator T: D(T) C X - Y we are dealing here with, the adjoint operator T* need not exist. But, following Kato [7], we can introduce an operator T+ which is essentially equivalent to the adjoint and which is defined as follows. Let X,, be the closure of D(T) in X and let T,, be the trivial restriction of T to X0 . Then X0 is a Banach space and T,, : D( TO) = D(T) C X0 --f Y is a closed linear operator with domain D( T,,) which is dense in X0. Hence the adjoint T,,*: D(T,*) C Y* + X,,* exists and we define T+ by setting TT = T,,*. Thus, T+ is a closed linear operator with domain D(T+) which is not necessarily dense in Y*. However, D(T+) is weakly dense in Y* in the sense that if y E Y and (y*, y} = 0 for all y* E D(T+), then y = 0. In what follows we shall use the following result from [7].

PROPOSITION 1. Suppose T: D(T) C X---f Y is closed with R(T) closed. Then TX = y has a solution x E D(T) if and only if y E N(T+)I. Moreover, N(T+) = R(T)l.

Following Kuratowski [9] we define y(Q), the set-measure of noncompactness of a bounded set Q C X, to be the inf{d > 0 [ Q can be covered by a finite number of sets of diameter < d}. It follows immediately that y(Q) = 0 if and only if Q is precompact. Closely associated with y is the notion of a k-set- contraction defined to be a continuous bounded mapping, say, A: G C X --) I7 such that y(A(Q)) < Ky(Q) for each bounded subset Q of G and some fixed R > 0, where for notational simplicity we use y to denote the set-measure of noncompactness both in X and in Y. It follows that C: G C X + Y is compact if and only if C is 0-set-contractive. If F: G C X- Y is I- Lipschitzian, then F and F + C are both k-set-contractive with k = 1. Following Sadovskii [14] (see also [2]) we say that A: G C X+ Y is set- condensing if r(A(Q)) quasinorm I A 1 if I A I = lim sup(lI 4Wll x II>< ~0. (2) IIXl!-+= UNBOUNDED LINEAR AND QUASIBOUNDED NONLINEAR OPERATORS 497

This concept has been introduced by Granas [4] who has shown that if A: X -+ X is a quasibounded compact map with 1 A ( < 1, then (I - A)(X) = X. A continuous map A: X --f Y is called asymptotically linear if there is A, eL(X, Y), called the asymptotic deriwatiwe of A, such that

,;,pm{II 4-4 - 44II/II x II> = 0. (3)

This notion has been introduced and used by Krasnoselskii [8] and earlier by Dubrovskii [l] for the case when (3) holds with A, = 0, i.e., when A is asymptotically zero. The following known result will prove to be useful (see [4, 81).

PROPOSITION 2. (i) If A: X --, Y is quasibounded, then to each E > 0 there corresponds Y > 0 such that j/ A(x)\1 < (I A j + E) I/ x/j for all x E X with I/ x 11> Y. (ii) If there exist constants a > 0 and b 3 0 such that /! iz(x)\i < a (/ x (1+ b for aZEx E X, then A is quasibounded with / A 1 < a. (iii) If A, EL(X, Y) is the asymptotic derivative of A, then A is quasibounded with 1A / = //A, 11.

We are now in the position to state our main result.

THEOREM 1. Let T: D(T) C X ---f Y be a closed linear mapping with D(T) not necessarily dense in X and such that R(T) is closed in Y and N(T) has a closed complementary subspace, say, X1 in X. Let S: X+ Y be a quasibounded mapping such that

(a) S(x) E N( T+)l for alE x E X. (b) T;lS, : X,---f X1 is 1-set-contractive and such that if (xn} C X, is any bounded sequence for which x, - T;lS,(x,) jg for some g in Xl , then there exists x E X1 such that x - T;lS,(x) = g, where T,(x) = T(x) for x E D(T) n X, and S, is the restriction of S to Xl . (c) 1 5’ 1 c < 1, where c is the norm of T;’ and 1 S / is the quasinorm of S. Then Eq. (l), TX + s(x) = Y, has a solution x E D(T) for a given y in Y if and only if y E N( T+)I.

Proof. Necessity. Since R(T) is closed in Y, Proposition 1 implies that N(T+)‘- = R(T) and so it follows from condition (a) that R(S) C R(T). Thus, if Eq. (1) has a solution x in D(T) for a given y in Y, then y E R(T). Hence, under condition (a), the assumption that y E N(T+)’ is necessary for Eq. (I) to have a solution in D(T). 498 W. V. PETRYSIIYN

Sufficiency. To prove the sufficiency note first that, by our hypothesis on N(T), there exists a closed subspace X, of X such that X = Ar( T) ~\z .X1 . Let Yr = R(T), D, == D( 7’) n Xi and define the operator Tl : D, C &Yl--+ 1, by T,(x) =z T(x) for .r: E D( TJ = D, , where X1 and Yr are considered as Banach spaces with respect to the norms inherited from X and 1’ respectively. It follows that Tl is an injective closed linear map from D(T,) onto 1, . Hence T; I: I; --f X1 exists and is a closed linear map defined on the complete space E; . Therefore, by the closed graph theorem, T;l is bounded and such that TT~T,(x) = .2*for x E D(T,) and TT;‘(y) = TIT;‘(y) = y for y E 1-r . We refer to T; ’ as a partiuE inverse of T and denote its norm by r. Now let y be an arbitrary but fixed element in N( T+)’ and let Sr, : X1 -+ 1, be defined by S,(X) = -S,(x) + y for x E X, . Since N(T+)mL = R(T), condition (a) and the above discussion imply that S,,(x) E R(T) for each x E X1 and that Eq. (1) has a solution x E D( TJ C D(T) if and only if the mapping P, - T;lS,, : X, + X1 has a fixed point in X1 . In view of condition (b), to show thatF, has a fixed point in Xi it suffices, by the author’s fixed point Theorem 1 in [13], to establish the existence of a ball B(0, r,) in Xi such that F,(x) # Xx for x E a&O, rV) and X > 1. Since, for any x E Xi and a > 1 we have the inequality

it follows from condition (c) and Lemma 1 (i) that for any E > 0 such that (I S 1 -+ E)C < 1 we may choose an r > 0 such that /I S(x)11 < (1 S / + c) 11x !I for all s E _‘i with jl x I! > Y. Thus, by choosing rV > max(r, c jl y Ii/(1 - (I S / + c)c) me see from the above inequality that for all x E aZ3(0, .y,) and all a: > I we have

I/ Fg(x) - ax 113 (I - c( I S I f c)) r - c i/y 11> 0.

Consequently, by Theorem 1 in [13], there exists x E B(0, r,) such that x = F,(x) == T;lS,,(x). Hence x E D(T,) C D(T) and, since Sly(x) E 77; , we see that T(x) == T(T;l S,,(x)) = S,,(.r) = S(x) - y, i.e., x is a solution of Eq. (1). Q.E.D.

Remark 1. Since N( T+) = iP( it follows that if N(P) = {0}, then R(T) = Y and so in this case condition (a) is superfluous. This is the case, in particular, when D(T) is dense in X and T*(=T+) is injective. Remark 2. Condition (b) holds if 7’~‘s~ : X1-X1 is set-condensing and, in particular, if S: X+ Y is k-set-contractive with ck < 1. Obviously, (b) holds when T;lSl is compact and the latter is true, in particular, when either S: S- I’ is compact or T;‘: Y, ---f ;Y, is compact and S is bounded. UNBOUNDED LINEAR AND QUASIBOUNDEDNONLINEAR OPERATORS 499

Remark 3. In view of Proposition 2, condition (c) holds if S is asymptotically linear with I] S, I] c < 1. The latter always holds if S is asymptotically zero since then 11S, jj = 0.

Remark 4. Theorem 1 remains valid if in (b) the condition that T;lS, : X,-+X, is I-set-contractive is replaced by the requirement that T;lS, is I-ball-contractive. (See [13, 141 for the definition of the ball-measure of noncompactness, k-ball-contractive, ball-condensing maps, and their properties.)

2. SPECIAL CASES

To illustrate the generality of Theorem 1 we deduce from it a couple of results which should prove to be useful in applications and at the same time will include as special cases the results of the various authors mentioned in the introduction. An immediate consequence of Theorem 1, Remarks 2 and 3, and Proposi- tion 2 is the following existence theorem for Eq. (1) involving compact perturbations.

THEOREM 2. Suppose T: D(T) C X---f Y is a closed linear map with D(T) not necessarily dense in X and such that R(T) is closed and N(T) has a closed complementary subspacein X. Let S: X+ Y be a compact map with the asymptotic deriwative S, GL(X, Y) and such that R(S) C R(T). (i) If S, = 0, then Eq. (1) h as a solution x E D(T) if and only ifv E R(T). (ii) If S, # 0 and I/ S, /j c < 1, then Eq. (1) has a solution x E D(T) if and only ify E R(T).

Remark 5. It is obvious that Theorem K follows from Theorem 2(i) since every finite dimensional subspace in X has a closed complementary subspace in X. If D(T) is dense in X and T: D(T) C X + Y admits a regularization (i.e., there exists B E L( Y, X) such that BT = I + L with I the identity and L compact on X) then, as was shown by Mikhlin [lo], R(T) is closed and dim N(T) < co. Hence Theorem 3 in [6] follows from Theorem 2(i). If F: X4 X is a compact map with the asymptotic derivative F, E L(X, X), then F(x) = F,(x) + N( x ) with F, compact (see [S]) and N compact and asymptotically zero. Hence Theorem 1 in [5] for the equation (I-F)(x) = y a1 so follows from Theorem 2(i) with T = I -F, and S = N. It has been suggested by Kachurovskii that the proof of his stated results utilizes Lemma 1 in [5] on normal solvability of a finite quasilinear 500 u'. V. PETRYSHYN

system of algebraic equations. This suggests that the finite dimensionality of N(T) assumed in [5, 61 appears to be essential for the indicated proof. Let me add that the surjectivity result of Dubrovskii [l] and Krasnoselskii [8] for asymptotically linear compact mappings follow from Theorem 2. A second consequence of Theorem 1, Remark 2 and Proposition 2 is the following result which includes the surjectivity theorems obtained in [4, 3, 12, 151 by different arguments.

THEOREM 3. Suppose T: D(T) C X -+ Y is a closed linear map with D(T) not necessarily dense in X such that N(T) = {0} and R(T) = Y. Let S: X-t Y be a quasibounded map such that condition (b) of Theorem 1 holds with X, = X, T;l = T-l and S, = S: (i) If 1S 1 = 0, then Eq, (1) has a soZution in D(T) for each y E Y. (ii) If 1S 1 # 0 and / S 1c < 1, then Eq. (1) has a solution in D(T) for each y in Y.

Remark 6. Since, as was noted in Remark 2, condition (b) is trivially satisfied when T-l is compact and since / S 1 = 0 when S is asymptotically zero, Theorem 2 of George [3] follows from Theorem 3(i). The surjectivity theorem of the author (see Corollary 2 in [12]) for l-set-contractions S: X -+ X with 1 S 1 < I and (I - S)(B(O, r)) closed for each Y > 0, as well as the earlier theorems of Granas [4] for S compact and of Vignoli [15] for S condensing follow from our Theorem 3(ii) when in the latter we set Y = X, D(T) = X and T = I and observe that in either of these cases condition (b) holds for 1 S 1 < 1 since c = 1. The applications of Theorems 1 and 2 to the solvability of certain differential and integro-differential equations whose linear and nonlinear parts satisfy the conditions of the above theorems will be discussed elsewhere.

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